$(30$ pts$)$ For this problem, refer to the Cellular Automata Handout and related PowerPoint slides.

$($a$)$ Convert the decimal number $240$ to a base$-2$ number. Use the binary $8-$digit representation of $240$

to draw a diagram like the one in Figure $1\mathrm{.}2$ in the Cellular Automata Handout for Rule $240.$

$($b$)$ Referencing your diagram from Part $($a$),$ create a space$-$time diagram for Rule $240,$ starting with

the same initial configuration as in Figure $1\mathrm{.}3$ in the Cellular Automata Handout and similarly

displaying four generations. Whereas Figure $1\mathrm{.}3$ is stated to have a finite initial configuration,

develop yours assuming the initial configuration is vec$((17))-$periodic $($the right neighbor of the

rightmost cell is the leftmost cell and vice versa$).$

$($c$)$ Consider a $1-$dimensional finite configuration $c$ containing the pattern $"111111"($six contiguous

cells in State $1),$ with all other cells in State $0$ and the leftmost State $1$ cell located at vec$(0).$ Describe

the set of configurations that map to configuration $c$ under the application of Rule $240,$ restricted

to having all cells in the position vec$(0)$ and to the left of it in State $0.$ What is the cardinality of this

set?